## Circuit Double Cover of GraphsThe famous Circuit Double Cover conjecture (and its numerous variants) is considered one of the major open problems in graph theory owing to its close relationship with topological graph theory, integer flow theory, graph coloring and the structure of snarks. It is easy to state: every 2-connected graph has a family of circuits covering every edge precisely twice. C.-Q. Zhang provides an up-to-date overview of the subject containing all of the techniques, methods and results developed to help solve the conjecture since the first publication of the subject in the 1940s. It is a useful survey for researchers already working on the problem and a fitting introduction for those just entering the field. The end-of-chapter exercises have been designed to challenge readers at every level and hints are provided in an appendix. |

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### Contents

Faithful circuit cover | 10 |

Circuit chain and Petersen minor | 21 |

Small oddness | 35 |

Spanning minor Kotzig frames | 45 |

Strong circuit double cover | 66 |

Spanning trees supereulerian graphs | 83 |

Flows and circuit covers | 96 |

Girth embedding small cover | 112 |

Shortest cycle covers | 163 |

Beyond integer 12weight | 189 |

Petersen chain and Hamilton weights | 199 |

Appendix A Preliminary | 243 |

Appendix B Snarks Petersen graph | 252 |

Integer flow theory | 273 |

Hints for exercises | 285 |

Glossary of terms and symbols | 322 |

Compatible circuit decompositions | 117 |

Other circuit decompositions | 134 |

Orientable cover | 153 |

References | 337 |

351 | |

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### Common terms and phrases

2-cell embedding 2-connected 2-factor 3-even subgraph cover 5-even subgraph double admits a nowhere-zero bridgeless cubic graph bridgeless graph CDC conjecture circuit decomposition circuit double cover circuit of G coloring component contradicts corollary cover F cover of G cover problem Deﬁnition denoted double cover conjecture edge e0 edge of G edge-cut of G edge-disjoint eulerian graph faithful circuit cover faithful cover faithful even subgraph Fano plane Figure Fleischner forbidden system G contains graph G graph G admits graph obtained Graph Theory Hamilton circuit Hamilton path Hint for Exercise induced path integer flow Jaeger Kotzig graph L-graph Lemma Let F Let G Let H members of F non-trivial nowhere-zero 4-flow obtained from G perfect matching permutation graph Petersen graph planar graph Proposition removable circuit semi-Kotzig snark spanning even subgraph subgraph double cover subgraph of G suppressed cubic graph Theorem Tutte uniquely 3-edge-colorable vertices weight of G weighted graph Zhang